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Euler-Rodrigues and Cayley formulas for rotation of

elasticity tensors

A. N. Norris

Mechanical and Aerospace Engineering, Rutgers University,

Piscataway NJ 08854-8058, USA norris@rutgers.eduDedicated to Professor Michael Hayes on the occasion of his 65th birthday

Abstract

It is fairly well known that rotation in three dimensions can be expressed as a quadraticin a skew symmetric matrix via the Euler-Rodrigues formula. A generalized Euler-Rodriguespolynomial of degree 2n in a skew symmetric generating matrix is derived for the rotationmatrix of tensors of order n. The Euler-Rodrigues formula for rigid body rotation is recoveredby n = 1. A Cayley form of the nth order rotation tensor is also derived. The representationssimplify if there exists some underlying symmetry, as is the case for elasticity tensors suchas strain and the fourth order tensor of elastic moduli. A new formula is presented for thetransformation of elastic moduli under rotation: as a 21-vector with a rotation matrix given bya polynomial of degree 8. Explicit spectral representations are constructed from three vectors:the axis of rotation and two orthogonal bivectors. The tensor rotation formulae are related toCartan decomposition of elastic moduli and projection onto hexagonal symmetry.

1 Introduction

Rigid body rotation about an axis p, |p| = 1, is described by the well known Euler-Rodriguesformula for the rotation matrix as a quadratic in the skew symmetric matrix P,

Pij = −ǫijk pk, (1)

where ǫijk is the third order isotropic alternating tensor1. Thus,

Q = exp(θP) = I + sin θ P + (1 − cos θ)P2, (2)

where θ is the angle of rotation. Euler first derived this formula although Rodrigues (21) obtainedthe formula for the composition of successive finite rotations (11; 7). The concise form of theEuler-Rodrigues formula is basically a consequence of the property

P3 = −P. (3)

1The summation convention on repeated subscripts is assumed in (1)1 and most elsewhere.

1

2

Hence, every term in the power series expansion of exp(θP) is either P2k = (−1)k+1P2 or P2k+1 =(−1)kP, and regrouping yields (2). Rotation about an axis is defined by the proper orthogonal rigidbody rotation matrix Q(θ,p) ∈ SO(3), the special orthogonal group of matrices satisfying2

QQt = QtQ = I, (4)

and the special property det Q = 1. The generating matrix P(p) ∈ so(3) where so(n) denotesthe space of skew symmetric matrices or tensors, usually associated with the Lie algebra of theinfinitesimal (or Lie) transformation group defined by rotations in SO(n). Alternatively, SO(3) isisomorphic to so(3) via the Cayley form (4)

Q = (I + S) (I− S)−1 = (I − S)−1 (I + S), (5)

where S ∈ so(3) follows from (2) as

S = tanh( θ2P) = tanθ

2P . (6)

The first identity is a simple restatement of the definition in (5) with Q = exp(θP), while thesecond is a more subtle relation that depends upon (3) and the skew symmetry of P, and as suchis analogous to the Euler-Rodrigues formula.

We are concerned with generalizing these fundamental formulas for rigid body rotation to therotation matrices associated with Q but which act on second and higher order tensors in thesame way that Q transforms a vector as v → v′ = Qtv. If one takes the view that the rotationtransforms a set of orthonormal basis vectors e1, e2, e3, then the coordinates of a fixed vectorv = v1e1 + v2e2 + v3e3 relative to this basis transform according to (v1, v2, v3) → (v′

1, v′2, v

′3), where

v′i = Qijvj . A tensor T of order (or rank) n has elements, or components relative to a vector basis,

Tij...kl. Under rotation of the basis vectors by Q, the elements are transformed by the relations

T ′ij...kl = Qij...klpq...rs Tpq...rs, (7)

where Q = Q(θ,p) is a tensor of order 2n formed from n combinations of the underlying rotation,

Qij . . . kl︸ ︷︷ ︸n

pq . . . rs︸ ︷︷ ︸n

≡ QipQjq . . . QkrQls︸ ︷︷ ︸n

. (8)

Q is also known as the n-th Kronecker power of Q (18). Lu and Papadopoulos (14) derived ageneralized n-th order Euler-Rodrigues formula for Q as a polynomial of degree 2n in the tensorP, also of order 2n, defined by the sum of n terms

Pij...klpq...rs = Pipδjq . . . δkrδls + δipPjq . . . δkrδls + . . . + δipδjq . . . Pkrδls + δipδjq . . . δkrPls . (9)

Thus,Q = exp(θ P) = P2n(θ, P), (10)

where

P2n(θ, x) =n∑

k=−n

eλkθ

n∏

j=−nj 6=k

( x − λj

λk − λj

), with λk = k i, i =

√−1 . (11)

Lu and Papadopoulos’ derivation of (11) is summarized in Section 3. The case n = 2 has also beenderived in quite different ways by Podio-Guidugli and Virga (20) and by Mehrabadi et al. (16).

2Qt denotes transpose and I is the identity.

3

The purpose of this paper is twofold. First, to obtain alternative forms of the generalizedEuler-Rodrigues formula and of the associated Cayley form. Using general properties of skewsymmetric matrices we will show that the n-th order Euler-Rodrigues formula can be expressed inways which clearly generalize the classical n = 1 case. The difference in approach from that of Luand Papadopoulos (14) can be appreciated by noting that their formula (11) follows from the factthat the eigenvalues of P are λk, k = −n, . . . , n, of eq. (11). The present formulation is based onthe more fundamental property that P has canonical form

P = P1 + 2 P2 + . . . + n Pn, (12)

where Pj satisfy identities similar to (3), and are mutually orthogonal in the sense of tensor product.This allows us to express exp(θ P) and tanh( θ

2P) in forms that more clearly generalize (2) and (6).

Furthermore, the base tensors Pj can be expressed in terms of powers of P, which imply explicitequations for Q and the related S. We will derive these equations in Section 3.

The second objective is to derive related results for rotation of tensors in elasticity. The under-lying symmetry of the physical tensors, strain, elastic moduli, etc., allow further simplification fromthe 2n− tensors Q ∈ SO(3n) and P ∈ so(3n), where the nominal dimensionality 3n assumes nosymmetry. Thus, Mehrabadi et al. (16) expressed the rotation tensor associated with strain, n = 2,in term of second order tensors in 6-dimensions, significantly reducing the number of componentsrequired. By analogy, the rotation matrix for fourth order elastic stiffness tensors is derived hereas a second order tensor in 21-dimensions Q(θ,p) = exp

(θP

)∈ SO(21) where P ∈ so(21). This

provides an alternative to existing methods for calculating elastic moduli under a change of basis,e.g. using Bond transformation matrices (3) or other methods based on representations of the mod-uli as elements of 6 × 6 symmetric matrices (16). Viewing the elastic moduli as a 21-vector is thesimplest approach for some purposes, such as projection onto particular symmetries (12; 6). In fact,we will see that the generalized Euler-Rodrigues formula leads to a natural method for projectingonto hexagonal symmetry defined by the axis p, as an explicit expression for the projector appearsquite naturally.

The third and final result is a general formulation of the so-called Cartan decomposition (8) oftensor rotation. The action of Q of (2) on a vector leaves the component parallel to the axis p

unchanged, and the component perpendicular to it rotates through angle θ. The latter decreasesthe part perpendicular to p to cos θ of its original value, and introduces sin θ times the samemagnitude but in in a direction orthogonal to both the vector and the axis. This simple geometricalinterpretation generalizes for tensors, in that we can identify “components” that remain unchanged,and others that rotate according to cos jθ, sin jθ, j = 1, 2, . . . , n. The components form subspacesthat rotate independently of one another, just as the axial and perpendicular components of thevector do for n = 1. This is the essence of Cartan decomposition of tensors, see (8) for furtherdetails. Our main result is that the components and the subspaces can be easily identified anddefined using the properties of P and Q that follow from (12).

We begin with a summary of the main findings in Section 2, with general results applicable totensors of arbitrary order described in Section 3. Further details of the proofs are given in Section4. Applications to elasticity are described in Section 5 where the 21 dimensional rotation of elasticmoduli is derived. Finally, the relation between the generalized Euler-Rodrigues formula and theCartan decomposition of elasticity tensors is discussed in Section 6.

A note on notation: tensors of order n are denoted P, Q, etc. Vectors and matrices in 3D arenormally lower and capital boldface, e.g. p, Q, while quantities in 6-dimensions are denoted e.g.Q, and in 21-dimensions as Q.

4

2 Summary of results

Our principal result is

Theorem 1 The rotation tensor Q(θ,p) = exp(θ P

)has the form

Q = I +n∑

k=1

(sin kθ Pk + (1 − cos kθ) P2

k

)(13)

where Pk, k = 1, 2, . . . , n are mutually orthogonal skew-symmetric tensors that partition P and havethe same basic property as P in (3),

P = P1 + 2 P2 + . . . + n Pn, (14a)

Pi Pj = Pj Pi = 0, i 6= j, (14b)

P3i = −Pi . (14c)

The Cayley form for Q is

Q = ( I + S) ( I − S)−1 = ( I − S)−1 ( I + S), (15)

where

S =

n∑

k=1

tan(k θ2) Pk . (16)

The tensors Pk are uniquely defined by polynomials of P of degree 2n − 1,

Pk = pn,k( P), where pn,k(x) =x

k

n∏

j=1j 6=k

(x2 + j2

j2 − k2

), 1 ≤ k ≤ n. (17)

The polynomial representation Q = P2n(θ, P) has several alternative forms, three of which are asfollows:

P2n(θ, x) = 1 +

n∑

k=1

2(−1)k+1

(n − k)! (n + k)!

[k sin kθ x + (1 − cos kθ) x2

] n∏

j=1j 6=k

(x2 + j2) (18a)

= 1 +

n∑

k=1

(2 sin θ2)2k−1

k(2k − 1)!

(k cos θ

2x + sin θ

2x2)

k−1∏

l=1

(x2 + l2) (18b)

= 1 + sin θ x(1 +

X1

1

(1 +

X2

2

(1 +

X3

3(1 + . . .

Xn−1

n − 1))))

+ (1 − cos θ) x2(1 +

X1

2

(1 +

X2

3

(1 +

X3

4(1 + . . .

Xn−1

n))))

, (18c)

whereXj = (1 − cos θ)(x2 + j2)/(2j + 1). (19)

We note that n = 1, 2, 3, 4 corresponds to rotation of vectors, second, third and fourth ordertensors, respectively. All these are covered by considering n = 4 in Theorem 1, for which the series(18c) and (18a) are, respectively,

P8(θ, x) =1 + sin θ x

(1 +

(1 − cos θ)

1.3(x2 + 1)

[1 +

(1 − cos θ)

2.5(x2 + 4)

(1 +

(1 − cos θ)

3.7(x2 + 9)

)])

5

+ (1 − cos θ) x2

(1 +

(1 − cos θ)

2.3(x2 + 1)

[1 +

(1 − cos θ)

3.5(x2 + 4)

(1 +

(1 − cos θ)

4.7(x2 + 9)

)])

= 1 +1

5!3

[sin θ x + (1 − cos θ)x2

](x2 + 4)(x2 + 9)(x2 + 16)

− 1

6!

[2 sin 2θ x + (1 − cos 2θ)x2

](x2 + 1)(x2 + 9)(x2 + 16)

+2

7!

[3 sin 3θ x + (1 − cos 3θ)x2

](x2 + 1)(x2 + 4)(x2 + 16)

− 2

8!

[4 sin 4θ x + (1 − cos 4θ)x2

](x2 + 1)(x2 + 4)(x2 + 9) . (20)

This expression includes all the Euler-Rodrigues type formulas for tensors of order n ≤ 4, on accountof the characteristic polynomial equation for P of degree 2n + 1,

P( P2 + I)( P2 + 4 I) . . . ( P2 + n2 I) = 0 . (21)

Thus, the last line in (20) vanishes for n = 3, the last two for n = 2, and all but the first line forn = 1, while the terms that remain can be simplified using (21). The formulas for P2n, n = 2 and3, are

P4(θ, x) =1 + sin θ x(1 +

1

3(1 − cos θ)(x2 + 1)

)+ (1 − cos θ) x2

(1 +

1

6(1 − cos θ)(x2 + 1)

), (22a)

P6(θ, x) =1 + sin θ x(1 +

1

3(1 − cos θ)(x2 + 1)

[1 +

1

10(1 − cos θ)(x2 + 4)

])

+ (1 − cos θ) x2(1 +

1

6(1 − cos θ)(x2 + 1)

[1 +

1

15(1 − cos θ)(x2 + 4)

]). (22b)

These identities apply for n ≤ 2 and n ≤ 3, respectively, since they reduce to, e.g. the n = 1formula using (21).

The second main result is a matrix representation for simplified forms of P and Q for 4th orderelasticity tensors. Although of fourth order, symmetries reduce the maximum number of elementsfrom 34 to 21, and hence the moduli can be described by a 21-vector and P and Q by 21 × 21matrices. This reduction in matrix size is similar to the 6×6 representation of Mehrabadi et al. (16)for symmetric second order tensors. The case of third order tensors (n = 3) is not discussed hereto the same degree of detail as for second and fourth order, but it could be considered in the samemanner. The following Theorem provides the matrices for n = 1, 2, 4, corresponding to rotation ofvectors, symmetric second order tensors, and fourth order elasticity tensors, respectively.

Theorem 2 The skew-symmetric matrices and the associated rotation matrices defined by the ten-sors ( P, Q) for n = 1, 2, 4, are given by (P,Q), (P, Q) and (P, Q), respectively. The m × m,m = 3, 6, 21, skew symmetric generator matrix for each case has the form

P = R − Rt, (23)

where

R = −X, for vectors, n = 1, (24a)

R =

(0

√2Y√

2Z X

), for symmetric tensors, n = 2, (24b)

6

R =

0 0 0 0 2Y 0 0

0 0 0 −√

2Y 0√

2N 0

0 0 0 0 0 2N −√

2Y

0 −√

2Z 0 0 X 0 −√

2X

0√

2N 2N 0 0 X 0

2Z 0 0 X 0 0√

2X

0 0 −√

2Z −√

2X√

2X 0 −X

,for elasticity

tensors, n = 4,

(24c)

with 0 = 03×3 and

X =

0 p3 00 0 p1

p2 0 0

, Y =

0 p2 00 0 p3

p1 0 0

, Z =

0 p1 00 0 p2

p3 0 0

, N =

p1 0 00 p2 00 0 p3

. (25)

In each case, the Q-matrix is given by Theorem 1, and the tensors acted on by Q are m-vectorswhich transform like v′ = Qv.

The specific form of the 6- and 21-vectors are given in Section 5.The third and final result identifies subspaces that are closed under rotation:

Theorem 3 The Cartan components of an n-th order tensor T are defined as

T0 = M0 T, Tj ≡ Mj T, Rj ≡ Pj T, j = 1, 2, . . . , n. (26)

where the n+1 symmetric projection tensors, Mk, k = 0, 1, . . . , n, are

M0 ≡ I + P21 + P2

2 + . . . + P2n, Mi = −P2

i , i = 1, 2, . . . , n. (27)

They satisfy, for 0 ≤ k ≤ n and 1 ≤ i ≤ n,

Mi Mk = Mk Mi = Mk Pi = Pi Mk = 0, i 6= k, (28a)

M2k = Mk, Mi Pi = Pi Mi = Pi, (28b)

Mk = mn,k

(P), mn,k(x) =

(−1)k (2 − δk0)

(n − k)!(n + k)!

n∏

j=0j 6=k

(x2 + j2). (28c)

The projection T0 along with the n−pairs {Tj, Rj}, j = 1, 2, . . . , n define n + 1 subspaces thatare closed under rotation:

Q T0 = T0, Q Tj = cos jθ Tj +sin jθ Rj , Q Rj = cos jθ Rj−sin jθ Tj, j = 1, 2, . . . , n. (29)

The meaning will become more apparent by example, as we consider the various cases of tensors oforder n = 1, 2, and 4 in Section 6.

7

3 General theory for tensors

3.1 P ’s and Q’s

The transpose of a 2n-th order tensor is defined by interchanging the first and last n indices. Inparticular, the skew symmetry of P and the definition of P in (9) implies that Pt = −P. Based onthe definitions of Q in (8) and the properties of the fundamental rotation Q ∈ SO(3), Q of (10)satisfies

d Q

dθ= P Q, (30)

where the product of two tensors of order 2n is another tensor of order 2n defined by contractingover n indices, (AB)ij...klpq...rs = Aij...klab...cd Bab...cdpq...rs. This gives meaning to the representation

Q = eθ P. (31)

Based on the skew symmetry of P it follows that

Q Qt = Qt Q = I, where Iij...klpq...rs =

n︷ ︸︸ ︷δipδjq . . . δkrδls . (32)

The isomorphism between the space of n-th order tensors and a vector space of dimension 3n impliesthat Q ∈ SO(3n) and P ∈ so(3n).

The derivation of the main result in Theorem 1 is outlined next using a series of Lemmas. Detailsof the proof are provided below and in the Appendix.

Lemma 1 For any N ≥ 1, a given non-zero N × N skew symmetric matrix B has 2m ≤ Ndistinct non-zero eigenvalues of the form {±ic1, ±ic2, . . . ,±icm}, c1, . . . , cm real, and B has therepresentation

B = c1B1 + c2B2 + . . . + cmBm, (33a)

BiBj = BjBi = 0, i 6= j, (33b)

B3i = −Bi. (33c)

Lemma 1 is essentially Theorem 2.2 of Gallier and Xu (9), who also noted the immediate corollary

eB = I +

∞∑

k=1

1

k!

(c1B1 + c2B2 + . . . + cmBm

)k

= I +∞∑

k=1

1

k!

(ck1B

k1 + ck

2Bk2 + . . . + ck

mBkm

)

= ec1B1 + ec2B2 + . . . + ecmBm − (m − 1)I

= I +

m∑

j=1

(sin cjBj + (1 − cos cj)B

2j

). (34)

Thus, the exponential of any skew symmetric matrix has the form of a sum of Euler-Rodriguesterms.

Lemma 2 The elements of the decomposition of Lemma 1 can be expressed in terms of the matrixB and its eigenvalues by

Bj =B

cj

m∏

k=1k 6=j

(B2 + c2k

c2k − c2

j

), (35)

8

Equation (35) may be obtained by starting with

B(B2 + c2k) =

m∑

j=1j 6=k

(c2k − c2

j )cjBj , (36)

and iterating until a single Bj remains. Equation (35) in combination with (34) and B2j = c−1

j BBj

implies that eB can be expressed as a polynomial of degree 2m in B,

eB = I +

m∑

j=1

[cj sin cj B + (1 − cos cj) B2

]c−2j

m∏

k=1k 6=j

(B2 + c2k

c2k − c2

j

). (37)

We are now ready to consider exp(θ P).

Lemma 3 The non-zero eigenvalues of the skew symmetric tensor P defined in eq. (9) are

{i, −i, 2i, −2i, . . . , ni,−ni}.

This follows from Zheng and Spence (23) or Lu and Papadopoulos (14), and is discussed in detailin Section 4. P therefore satisfies the characteristic equation (21), and has the properties

P = P1 + 2 P2 + . . . + n Pn, (38a)

Pi Pj = Pj Pi = 0, i 6= j, (38b)

P3i = −Pi . (38c)

Equation (38a) is the canonical decomposition of P into orthogonal components each of whichhas properties like the fundamental P in (3), although the subspace associated with Pi can bemultidimensional. The polynomials pn,k(x) of eq. (17) follow from (35), or alternatively,

pn,k(x) =(−1)k+1 2kx

(n − k)!(n + k)!

n∏

j=1j 6=k

(x2 + j2), 1 ≤ k ≤ n. (39)

We will examine the particular form of the Pi for elasticity tensors of order n = 2 and n = 4 inSection 5. For now we note the consequence of Lemmas 2 and 3,

eθ P = I +n∑

k=1

[k sin kθ P + (1 − cos kθ) P2

]k−2

n∏

j=1j 6=k

( P2 + j2 I

j2 − k2

). (40)

Together with (31) this implies the first expression (18a) in Theorem 1. The alternative identities(18b) and (18c) are derived in the Appendix.

Lemma 4 Q may be expressed in Cayley form

Q = ( I + S) ( I − S)−1 = ( I − S)−1 ( I + S), (41)

where the 2n-th order skew symmetric tensor S is

S = tanh( θ2

P) =

n∑

k=1

tan(k θ2) Pk . (42)

9

This follows by inverting (41) and using the expression (13) for Q,

S = I − 2( I + Q)−1 = I −[I +

n∑

k=1

1

2

(sin kθ Pk + (1 − cos kθ) P2

k

)]−1, (43)

and then applying the following identity,

[I +

n∑

k=1

(ak Pk + (1 − bk) P2

k

)]−1= I +

n∑

k=1

(− ak

a2k + b2

k

Pk + (1 − bk

a2k + b2

k

) P2k

). (44)

Note that Lemma 4 combined with the obvious result (42)1 implies that, for arbitrary φ,

tanh(φ P) =n∑

k=1

tan(kφ) Pk. (45)

This is a far more general statement than the simple partition of (12), and in fact may be shownto be equivalent to eq. (38).

3.2 Equivalence with the Lu and Papadopoulos formula

We now show that (40) agrees with the polynomial of Lu and Papadopoulos (14). They derived eq.(11) directly using Sylvester’s interpolation formula (p. 437 of Horn and Johnson (13)). Thus, thefunction f(A) defined by a power series for a matrix A that is diagonalizable can be expressed interms of its distinct eigenvalues λ1, λ2, . . . λr as f(A) = p(A), where p(x) is

p(x) =r∑

i=1

f(λi) Li(x), Li(x) =r∏

j=1j 6=i

( x − λj

λi − λj

). (46)

Li(x) are the Legendre interpolation polynomials and p(x) is the unique polynomial of degree r− 1with the property p(λi) = f(λi). Equation (11) follows from the explicit form of the spectrum ofP. The right member of (11) can be rewritten in the following form by combining the terms e±ikθ,

P2n(θ, x) =1

(n!)2

n∏

j=1

(x2 + j2) +n∑

k=1

2(−1)k+1

(n − k)! (n + k)!

(k sin kθ x − cos kθ x2

) n∏

j=1j 6=k

(x2 + j2) . (47)

The first term on the right hand side can be written in a form which agrees with (18a) by consideringthe partial fraction expansion of 1/Λ where

Λ(x) =

n∏

k=1

(x2 + k2) . (48)

Thus, as may be checked by comparing residues, for example,

1

Λ=

n∑

k=1

2(−1)k+1

(n − k)! (n + k)!

k2

x2 + k2

=n∑

k=1

2(−1)k+1

(n − k)! (n + k)!−

n∑

k=1

2(−1)k+1

(n − k)! (n + k)!

x2

x2 + k2. (49)

10

The first term in the right member is 1/Λ(0) = 1/(n!)2. Multiplying by Λ(x)/Λ(0) implies theidentity

1

(n!)2

n∏

j=1

(x2 + j2) = 1 +

n∑

k=1

2x2(−1)k+1

(n − k)! (n + k)!

n∏

j=1j 6=k

(x2 + j2) , (50)

which combined with (47) allows us recover the form (18a). This transformation from the interpo-lating polynomial (11) to the alternative forms in eq. (18) is rigorous but it does not capture thephysical basis of the latter. We will find the identity (50) useful in Section 6.

4 Eigenvalues of P and application to second order tensors

The key quantity in the polynomial representation of Q is the set of distinct non-zero eigenvaluesof P, Lemma 3. We prove this by construction, starting with the case of n = 1.

4.1 Rotation of vectors, n = 1

Let {p,q, r} form an orthonormal triad of vectors, then the eigenvalues and eigenvectors of P areas follows

eigenvalue eigenvector0 p

±i v± ≡ 1√2(iq ± r)

(51)

These may be checked using the properties of the third order alternating tensor. Thus, Pijqj = ri,Pijrj = −qi, implying Pv± = ±iv±. Hence, the spectral representation of P is

P = iv+v∗+ − iv−v∗

− , (52)

where ∗ denotes complex conjugate. It also denotes transpose if we view eq. (52) in vector/matrixformat. We can also think of this as a tensorial representation in which terms such as v−v∗

− standfor the Hermitian dyadic v−⊗v∗

−, however, for simplicity of notation we do not use dyadic notationfurther but take the view that dyadics are obvious from the context.

The bivectors (5) v±, together with the axis p, will serve as the building blocks for spectralrepresentations of P for n > 1.

Referring to eq. (38), we see that P = P1 in this case. Equation (52) allows us to evaluatepowers and other functions of P. Thus, in turn,

P2 = −v+v∗+ − v−v∗

− , (53a)

P2 + I = pp, (53b)

P(P2 + I) = 0, (53c)

from which the characteristic equation (3) follows.

4.2 Rotation of second order tensors, n = 2

In this case the tensors P and Q are fourth order with, see eq. (9),

Pijkl = Pikδjl + δikPjl . (54)

11

Equation (17) impliesP1 = P( P2 + 4)/3, P2 = −P( P2 + 1)/6, (55)

which clearly satisfy the decomposition (38a)

P = P1 + 2 P2. (56)

The skew symmetric tensor S of (16) may be expressed

S =(

43tan θ

2− 1

6tan 3θ

2

)P +

(13tan θ

2− 1

6tan 3θ

2

)P3. (57)

Consider the product of the second order tensor (dyad) pv± with P. Thus, Ppv± = ±ipv±which together with Pv±p = ±iv±p yields 4 eigenvectors, two pairs with eigenvalue i and −i.The dyadics v±v± are also eigenvectors and have eigenvalues ±2i. The remaining 3 eigenvectorsof P are null vectors which can be identified as pp, v+v− and v−v+. This completes the 3n = 9eigenvalues and eigenvectors, and shows that the non-zero eigenvalues are {i(2),−i(2), 2i,−2i}where the number in parenthesis indicates the multiplicity. The tensors P1 and P2 can be expressedexplicitly in terms of the eigenvectors,

P1 = ipv+pv∗+ + iv+pv∗

+p − ipv−pv∗− − iv−pv∗

−p, P2 = iv+v+v∗+v∗

+ − iv−v−v∗−v∗

−. (58)

It is straightforward to show that these satisfy the conditions (38).We note in passing that zero eigenvalues correspond to tensors of order n with rotational or

transversely isotropic symmetry about the p axis. This demonstrates that there are three secondorder basis tensors for transversely isotropic symmetry. We will discuss this aspect in greater detailin Section 6 in the context of elasticity tensors.

Both P and Q correspond to 3n × 3n or 9 × 9 matrices, which we denote P and Q, with theprecise form dependent on how we choose to represent second order tensors as 9-vectors. To bespecific, let Tij be the components of a second order tensor (not symmetric in general), and definethe 9-vector

T =(T11, T22, T33, T23, T31, T12, T32, T13, T21

)t. (59)

The form of P follows from the expansion (7) for small θ, or from (9), as

P =

0 0 0 0 p2 −p3 0 p2 −p3

0 0 0 −p1 0 p3 −p1 0 p3

0 0 0 p1 −p2 0 p1 −p2 00 p1 −p1 0 0 0 0 p3 −p2

−p2 0 p2 0 0 0 −p3 0 p1

p3 −p3 0 0 0 0 p2 −p1 00 p1 −p1 0 p3 −p2 0 0 0

−p2 0 p2 −p3 0 p1 0 0 0p3 −p3 0 p2 −p1 0 0 0 0

. (60)

This can be written in block matrix form using the X, Y, Z matrices defined in eq. (25),

P = R − Rt, where R =

0 Y Y

Z 0 X

Z X 0

. (61)

Note that this partition of the skew symmetric matrix is not unique.

12

4.3 Tensors of arbitrary order n ≥ 2

The example of second order tensors shows that the three eigenvectors {p, v+, v−} of the fun-damental matrix P define all 3n = 9 eigenvectors of P. This generalizes to arbitrary n ≥ 2 bygeneration of all possible n-tensors formed from the tensor outer product of {p, v+, v−}. For anyn, consider the product P with the n-tensor

v+ = v+v+ . . .v+︸ ︷︷ ︸n

. (62)

Each of the n terms of P in (9) contributes +iv+ with the result Pv+ = ni v+. Similarly, defining v−implies Pv− = −ni v−, from which it is clear that there are no other eigenvectors with eigenvalues±ni. Thus, the final component in the decomposition (38a) is

Pn = i v+v∗+ − i v−v∗

− . (63)

Next consider the n n-th order tensors that are combinations of (n − 1) times v+ with a singlep, e.g.

v = pv+v+ . . .v+︸ ︷︷ ︸n−1

. (64)

The action of P is Pv = (n−1)iv, and so there are 2n eigenvectors with eigenvalues ±(n−1)i whichcan be combined together to form the component Pn−1, as in eq. (58)1 for n = 2. Eigenvalues(n − 2)i are obtained by combining p twice with v+ (n − 2) times, but also come from n-tensorsformed by a single v− with (n − 1) times v+. Enumeration yields a total of n(n + 1) eigenvectorsassociated with Pn−2. By recursion, it is clear that the number of eigenvectors with eigenvalues kiequals the number of ways n elements chosen from {−1, 0, 1} sum to −n ≤ k ≤ n, or equivalently,the coefficient of xk in the expansion of

(1 + x + x−1

)n,

(1 + x + x−1

)n=

n∑

k=−n

(n

k

)

2

xk . (65)

The numbers(

n

k

)2

are trinomial coefficients (not to be confused with q-multinomial coefficients (1)

with q = 3) and can be expressed (2)

(n

k

)

2

=

n∑

j=0

(n

2j + k

)(2j + k

j

), (66)

where(

n

j

)= n!/(j!(n − j)!) are binomial coefficients. The main point for the present purpose is

thatn∑

k=−n

(n

k

)

2

= 3n, (67)

which is a consequence of (65) with x = 1. It is clear from this process that the entire set of 3n

eigenvectors has been obtained and there are no other candidates for eigenvalue of P. The preciseform of the eigenvectors is unimportant because the Euler-Rodrigues type formulas (18) dependonly on the set of distinct non-zero eigenvalues. However, the eigenvectors do play a role in settingup matrix representations of P for specific values of n, and in particular for tensors with symmetries,as discussed in Section 5.

13

5 Applications to elasticity

We first demonstrate that the underlying symmetry of second and fourth order tensors in elasticityimplies symmetries in the rotation tensors which reduces the number of independent elements inthe latter.

5.1 Rotation of symmetric second and fourth order tensors

We consider second and fourth order tensors such as the elastic strain ε (n = 2) and the elasticstiffness tensor C (n = 4) with elements εij and Cijkl. They transform as εij → ε′ij and Cijkl → C ′

ijkl,

ε′ij = Qijpqεpq C ′ijkl = Qijklpqrs Cpqrs , (68)

where the rotation tensors follow from the general definition (8) as

Qijpq = QipQjq, Qijklpqrs = QijpqQklrs . (69)

The strain and stiffness tensors possess physical symmetries which reduce the number of independentelements that need to be considered,

εij = εji, Cijkl = Cjikl = Cijlk, Cijkl = Cklij. (70)

In short, the number of independent elements of ε is reduced from 9 to 6, and of C from 81 to 21.Accordingly, we may define variants of Qijkl and Qijklpqrs with a reduced number of elements thatreflect the underlying symmetries of elasticity. Thus, the transformation rules can be expressed inthe alternative forms

ε′ij = Qijpqεpq, C ′ijkl = Qijklpqrs Cpqrs , (71)

where the elements of the symmetrized 4th and 8th order Q tensors are

Qijpq =1

2

(Qijpq + Qijqp

), (72a)

Qijklpqrs =1

2

(QijpqQklrs + QijrsQklpq

). (72b)

Alternatively, they can be expressed in terms of the fundamental Q ∈ SO(3),

Qijpq =1

2

(QipQjq + QiqQjp

), (73a)

Qijklpqrs =1

8

(QipQjqQkrQls + QipQjqQksQlr + QiqQjpQkrQls + QiqQjpQksQlr

+ QirQjsQkpQlq + QirQjsQkqQlp + QisQjrQkpQlq + QisQjrQkqQlp

). (73b)

Using the identity (32), we have

QijpqQklpq =1

2

(Iijkl + Iijlk

)= I ijkl ≡

1

2

(δikδjl + δilδjk

), (74)

which is the fourth order isotropic identity tensor, i.e. the fourth order tensor with the propertys = Is for all symmetric second order tensors s. Similarly,

QijklabcdQpqrsabcd =1

2

(I ijpqIklrs + I ijrsIklpq

)≡ I ijklpqrs . (75)

14

This is the 8th order isotropic tensor with the property C = I C for all 4th order elasticity tensors.In summary, the symmetrized rotation tensors for strain and elastic moduli are orthogonal in the

sense that Q Qt= I, and are therefore elements of SO(6) and SO(21), respectively.

The (symmetrized) rotation tensor for strain displays the following symmetries

Qijpq = Qjipq = Qijqp. (76)

Introducing the Voigt indices, which are capital suffices taking the values 1, 2, . . . , 6 according to

I = 1, 2, 3, 4, 5, 6 ⇔ ij = 11, 22, 33, 23, 31, 12, (77)

then (76) implies that the elements Qijpq can be represented by QIJ . Similarly, the elements of the

8th order tensor Qijklpqrs can be represented as QIJKL, which satisfy the symmetries

QIJKL = QJIKL = QIJLK . (78)

The pairs of indices IJ and KL each represent 21 independent values, suggesting the introductionof a new type of suffix which ranges from 1, 2, . . . , 21. Thus,

I = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21

⇔ IJ = 11, 22, 33, 23, 31, 12, 44, 55, 66, 14, 25, 36, 34, 15, 26, 24, 35, 16, 56, 64, 45. (79)

Hence the elements Qijklpqrs can be represented uniquely by the (21)2 elements QIJ . The elements

of the symmetrized rotations QIJ and QIJ have reduced dimensions, 6 and 21 respectively, but donot yet represent the matrix elements of 6- and 21-dimensional tensors. That step is completednext, after which we can define the associated skew symmetric generating matrices and return tothe generalized Euler-Rodrigues formulas for 6- and 21-dimensional matrices.

5.2 Six dimensional representation

We now use the isomorphism between second order tensors in three dimensions, such as the strainε, and six dimensional vectors according to ε → ε with elements εI , I = 1, 2, . . . , 6. Similarly,fourth order elasticity tensors in three dimensions are isomorphic with second order positive definitesymmetric tensors in six dimensions C with elements cIJ (15). Let {ε} be the 6-vector with elementsεI , I = 1, 2, . . . , 6, and [C] the 6 × 6 Voigt matrix of elastic moduli, i.e. with elements cIJ . Theassociated 6-dimensional vector and tensor are

ε = T{ε}, C = T[C]T, where T ≡ diag(1, 1, 1,

√2,

√2,

√2). (80)

Explicitly,

ε =

ε1

ε2

ε3

ε4

ε5

ε6

=

ε11

ε22

ε33√2ε23√2ε31√2ε12

, C =

c11 c12 c13 21

2 c14 21

2 c15 21

2 c16

c22 c23 21

2 c24 21

2 c25 21

2 c26

c33 21

2 c34 21

2 c35 21

2 c36

2c44 2c45 2c46

S Y M 2c55 2c56

2c66

. (81)

The√

2 terms ensure that products and norms are preserved, e.g. CijklCijkl = tr CtC.

15

The six-dimensional version of the fourth order tensor Qijkl is Q ∈ SO(6), introduced by

Mehrabadi et al. (16), see also (19). It may be defined in the same manner as Q = S[Q]S

where [Q] is the matrix of Voigt elements. Thus, QQt = QtQ = I, where I = diag(1, 1, 1, 1, 1, 1).

It can be expressed Q(p, θ) = exp(θP), and hence is given by the n = 2 Euler-Rodrigues formula

(22) for the skew symmetric P(p) ∈ so(6),

P =

0 0 0 0√

2p2 −√

2p3

0 0 0 −√

2p1 0√

2p3

0 0 0√

2p1 −√

2p2 0

0√

2p1 −√

2p1 0 p3 −p2

−√

2p2 0√

2p2 −p3 0 p1√2p3 −

√2p3 0 p2 −p1 0

, (82)

or in terms of the block matrices defined in (25),

P = R − Rt, R =

(0

√2Y√

2Z X

). (83)

It is useful to compare P with the 9×9 matrix for rotation of general second order tensors, eqs. (60)and (61). The reduced dimensions and the

√2 terms are a consequence of the underlying symmetry

of the tensors that are being rotated. Vectors and tensors transform as ε → ε′ and C → C′ where

ε′ = Q ε, C′ = QCQt . (84)

The eigenvalues and orthonormal eigenvectors of P are as follows

eigenvalue eigenvector dyadic

0 i 1√3I

0 pd

√32(pp− 1

3I)

±i u±1√2(pv± + v±p)

±2i v± v±v±

(85)

where v± are defined in (51) and

i ≡

1/√

3

1/√

3

1/√

3000

pd ≡√

3

2

p21 − 1

3

p22 − 1

3

p23 − 1

3√2 p2p3√2 p3p1√2 p1p2

, u± ≡

√2 p1v±,1√2 p2v±,2√2 p3v±,3

p2v±,3 + p3v±,2

p3v±,1 + p1v±,3

p1v±,2 + p2v±,1

, v± ≡

v2±,1

v2±,2

v2±,3√

2 v±,2v±,3√2 v±,3v±,1√2 v±,1v±,2

. (86)

The right column in (85) shows the second order tensors corresponding to the eigenvectors in dyadicform. The eigenvectors are orthonormal, i.e. of unit magnitude and mutually orthogonal. The unit6−vectors i and pd correspond to the hydrostatic and deviatoric parts of pp, respectively. Thedouble multiplicity of the zero eigenvalue means that any 6−vector of the form apd + bi is a null

vector of P, including√

23pd + 1√

3i which is the six-vector for the dyad pp.

16

The canonical decomposition of the generating matrix P ∈ so(6) is therefore

P = P1 + 2P2, P1 = iu+u∗+ − iu−u∗

−, P2 = iv+v∗+ − iv−v∗

− . (87)

The associated fourth order tensor P, which is the symmetrized version of P in eq. (56), is

P = P1 + 2 P2, P1 =∑

±

i

2(pv± + v±p)(pv∗

± + v∗±p), P2 =

∑

±±iv±v±v∗

±v∗±. (88)

The representation (87) allows us to compute powers of P, and using the spectral decompositionof the identity

I = iit + pdptd + u+u∗

+ + u−u∗− + v+v∗

+ + v−v∗− , (89)

the characteristic equation (16) follows:

P(P2 + I)(P2 + 4I) = 0 . (90)

5.3 21 dimensional representation

Vectors and matrices in 21-dimensions are denoted by a tilde. Thus, the 21-dimensional vector ofelastic moduli is c with elements cI , I = 1, 2, . . . , 21. Similarly, vectors corresponding to the fourthorder tensors εε and 1

2(A ⊗B + B ⊗ A) are ε and v, that is

4th order, n=3 2nd order, n=6 vector, n=21

C C c

εε εεt

ε

12(A ⊗ B + B ⊗ A) 1

2(abt + bat) v

. (91)

17

The 21 elements follow from the indexing scheme (79) and are defined by

c =

c1

c2

c3

c4

c5

c6

c7

c8

c9

c10

c11

c12

c13

c14

c15

c16

c17

c18

c19

c20

c21

=

c11

c22

c33√2c23√2c13√2c12

2c44

2c55

2c66

2c14

2c25

2c36

2c34

2c15

2c26

2c24

2c35

2c16

2√

2c56

2√

2c46

2√

2c45

=

c11

c22

c33√2c23√2c13√2c12

c44

c55

c66√2c14√2c25√2c36√2c34√2c15√2c26√2c24√2c35√2c16√2c56√2c46√2c45

, ε =

ε211

ε222

ε233√

2ε22ε33√2ε11ε33√2ε11ε22

2ε223

2ε213

2ε212

2ε11ε23

2ε22ε13

2ε33ε12

2ε33ε23

2ε11ε13

2ε22ε12

2ε22ε23

2ε33ε13

2ε11ε12

2√

2ε13ε12

2√

2ε23ε12

2√

2ε23ε13

=

ε21

ε22

ε23√

2ε2ε3√2ε3ε1√2ε1ε2

ε24

ε25

ε26√

2ε1ε4√2ε2ε5√2ε3ε6√2ε3ε4√2ε1ε5√2ε2ε6√2ε2ε4√2ε3ε5√2ε1ε6√2ε5ε6√2ε6ε4√2ε4ε5

, v =

a1b1

a2b2

a3b31√2(a2b3 + a3b2)

1√2(a3b1 + a1b3)

1√2(a1b2 + a2b1)

a4b4

a5b5

a6b61√2(a1b4 + a4b1)

1√2(a2b5 + a5b2)

1√2(a3b6 + a6b3)

1√2(a3b4 + a4b3)

1√2(a1b5 + a5b1)

1√2(a2b6 + a6b2)

1√2(a2b4 + a4b2)

1√2(a3b5 + a5b3)

1√2(a1b6 + a6b1)

1√2(a5b6 + a6b5)

1√2(a6b4 + a4b6)

1√2(a4b5 + a5b4)

.

(92)

Alternatively,c = T{c}, etc., (93)

where {c} is the 21×1 array with elements cI , and T is the the 21×21 diagonal with block structure

T = diag(I√

2I 2I 2I 2I 2I 2√

2I). (94)

For instance, the elastic energy density W = 12Cijklεijεkl is given by the vector inner product

W =1

2ε

tc , (95)

where ε is the 21-vector associated with strain (92).

5.3.1 The 21-dimensional rotation matrix

Vectors in 21-dimensions transform as c → c′ = Q c where Q ∈SO(21) satisfies QQt = QtQ = I,

and I is the identity, IIJ = δIJ . The rotation matrix Q can be expressed

Q(p, θ) = eθ eP, (96)

where P(p) ∈ so(21) follows from the general definition (9) applied to elasticity tensors, i.e. (73b).Thus,

P = T[P ]T, Q = T[Q]T (97)

18

where [P ], [Q] are the 21×21 array with elements P IJ , QIJ . We take a slightly different approach,

and derive the elements of P using the expansion of Q for small θ directly. Define the rotationalderivative of the moduli as

cijkl =dc′ijkl

dθ

∣∣∣∣θ=0

. (98)

Using (84)2 for instance, gives˙C = PC + CPt , (99)

from which we obtain

c11 = 4c15p2 − 4c16p3 , (100a)

c22 = −4c24p1 + 4c26p3 , (100b)

c33 = 4c34p1 − 4c35p2 , (100c)

c23 = 2(c24 − c34)p1 − 2c25p2 + 2c36p3 , (100d)

c13 = 2c14p1 + 2(c35 − c15)p2 − 2c36p3 , (100e)

c12 = −2c14p1 + 2c25p2 + 2(c16 − c26)p3 , (100f)

c44 = 2(c24 − c34)p1 − 2c46p2 + 2c45p3 , (100g)

c55 = 2c56p1 + 2(c35 − c15)p2 − 2c45p3 , (100h)

c66 = −2c56p1 + 2c46p2 + 2(c16 − c26)p3 , (100i)

c14 = (c12 − c13)p1 − (c16 − 2c45)p2 + (c15 − 2c46)p3 , (100j)

c25 = (c26 − 2c45)p1 + (c23 − c12)p2 − (c24 − 2c56)p3 , (100k)

c36 = −(c35 − 2c46)p1 + (c34 − 2c56)p2 + (c13 − c23)p3 , (100l)

c34 = −(c33 − c23 − 2c44)p1 − (c36 + 2c45)p2 + c35p3 , (100m)

c15 = c16p1 − (c11 − c13 − 2c55)p2 − (c14 + 2c56)p3 , (100n)

c26 = −(c25 + 2c46)p1 + c24p2 − (c22 − c12 − 2c66)p3 , (100o)

c24 = (c22 − c23 − 2c44)p1 − c26p2 + (c25 + 2c46)p3 , (100p)

c35 = (c36 + 2c45)p1 + (c33 − c13 − 2c55)p2 − c34p3 , (100q)

c16 = −c15p1 + (c14 + 2c56)p2 + (c11 − c12 − 2c66)p3 , (100r)

c56 = (c66 − c55)p1 + (c36 − c16 + c45)p2 − (c25 − c15 + c46)p3 , (100s)

c46 = −(c36 − c26 + c45)p1 + (c44 − c66)p2 + (c14 − c24 + c56)p3 , (100t)

c45 = (c25 − c35 + c46)p1 − (c14 − c34 + c56)p2 + (c55 − c44)p3 . (100u)

The elements of P can be read off from this and are given in Table 1. The 21 × 21 format can besimplified to a 7 × 7 array of block elements, comprising combinations of the 3 × 3 zero matrix 0

and the 3 × 3 matrices defined in (25). Thus,

P = R− Rt, where R =

0 0 0 0 2Y 0 0

0 0 0 −√

2Y 0√

2N 0

0 0 0 0 0 2N −√

2Y

0 −√

2Z 0 0 X 0 −√

2X

0√

2N 2N 0 0 X 0

2Z 0 0 X 0 0√

2X

0 0 −√

2Z −√

2X√

2X 0 −X

. (101)

19

Equation (24) shows that the rotation matrices in three, six and 21 dimensions can be simplifiedusing the fundamental 3 × 3 matrices X,Y,Z and N. The partition of a skew symmetric matrixin this way is not unique, but it simplifies the calculation and numerical implementation. Thesematrices satisfy several identities,

XXt + YYt + ZZt = I , (102a)

XtX = YYt, YtY = ZZt, ZtZ = XXt, (102b)

XtY = YZt, YtZ = ZXt, ZtX = XYt, (102c)

‖N‖ = ‖X‖ = ‖Y‖ = ‖Z‖ = 1, (102d)

where the norm is ‖u‖ =√

trut u .

We find that the eigenvalues of P are 0(5), i(3), −i(3), 2i(3), −2i(3), 3i, −3i, 4i, −4i, where

the number in parenthesis is the multiplicity. The associated orthonormal eigenvectors of P are

eigenvalue eigenvector 6-dyadic

0 v0a iit

0 v0b32pdp

td

0 v0c

√3

2(ipt

d + pdit)

0 v0d1√2(u−ut

+ + u+ut−)

0 v0e1√2(v−vt

+ + v+vt−)

±i v1a±1√2(iut

± + u± it)

±i v1b±√

32

(pdut± + u±pt

d)±i v1c±

1√2(v±ut

∓ + u∓vt±)

±2i v2a± u±ut±

±2i v2b±1√2(ivt

± + v±it)

±2i v2c±√

32

(pdvt± + v±p

td)

±3i v3±1√2(u±vt

± + v±ut±)

±4i v4± v±vt±

(103)

Apart from the last two, which have unit multiplicity, these are not unique. For instance, differentcombinations of the eigenvectors with multiplicity greater than one could be used instead. We alsonote from their definitions in eqs. (51) and (85) that

v∗± = −v∓, u∗

± = −u∓, v∗± = v∓. (104)

Hence, the five null vectors of P in (103) are real. P can be expressed in terms of the remaining 16eigenvectors as

P = P1 + 2P2 + 3P3 + 4P4, (105)

wherePj =

∑

±±i

∑

x=a,b,c

vjx±v∗jx±, j = 1, 2; Pj = vj±v∗

j±, j = 3, 4. (106)

The identity is

I =∑

x=a,b,c,d,e

v0xv0x +∑

±

( ∑

x=a,b,c

(v1x±v∗

1x± + v2x±v∗2x±

)+ v3±v∗

3± + v4±v∗4±

). (107)

20

It is straightforward to demonstrate using these expressions that P satisfies

P (P2 + 1) (P2 + 4) (P2 + 9) (P2 + 16) = 0 , (108)

which is the characteristic equation for 2n-th order P tensors, eq. (21).

6 Projection onto TI symmetry and Cartan decomposition

It has been mentioned in passing that null vectors of P and its particular realizations for elasticitytensors correspond to tensors with transversely isotropic symmetry. These are defined as tensorsinvariant under the action of the group SO(2) associated with rotation about the axis p. Weconclude by making this connection more specific, and relating the theory for the rotation of tensorsto the Cartan decomposition, which is introduced below.

6.1 Cartan decomposition

Referring to Theorem 3, the expansion of the projector Mk defined in eq. (27) as an even polynomialof degree 2n in P, i.e. of degree n in P2, follows from eqs. (39) and (50) and the identity P2

k =k−1 P Pk. Note that the Mk tensors partition the identity

I = M0 + M1 + . . . + Mn. (109)

Starting with M0 we haveP M0 = M0 P = 0, (110)

indicating that M0 is the linear projection operator onto the null space of P. In order to examinethe remaining n projectors, note that the rotation can be expressed

Q = M0 +

n∑

j=1

(cos jθ Mj + sin jθ Pj

). (111)

The action of Q on an n-th order tensor T, the rotation of T, can therefore be described as

Q T = T0 +

n∑

j=1

(cos jθ Tj + sin jθ Rj

), (112)

where T0 and Tj , Rj, j = 1, 2, . . . , n are defined by (26). We note that Rj may be expressed

Rj = Pj Tj =1

jP Tj , j = 1, 2, . . . , n. (113)

Also, T0 is the component of T in the null space of P, i.e. the transversely isotropic part of T.Equation (112) show that T0 and the pairs Tj , Rj , j=1,2,. . . , n, rotate separately, forming distinctsubspaces, and hence proving Theorem 3.

6.2 Application to elasticity tensors

The projection operators are now illustrated with particular application to tensors of low order andelasticity tensors, starting with the simplest case: rotation of vectors.

21

6.2.1 Vectors, n=1

If n = 1, we have T = P of eq. (1), and hence

M0 = pp, M1 = I − pp. (114)

Also, T → t, an arbitrary vector, and the decomposition (112) and (26) is

Qt = (t · p)p + cos θ [t − (t · p)p] + sin θ p×t. (115)

Thus, we identify t0 = M0t as the component of the vector parallel to the axis of rotation, t1 = M1t

as the orthogonal complement, and r1 = P1t is the rotation of t1 about the axis by π/2.

6.2.2 Second order tensors, n=2

For n = 2, we have P1, P2 given in (55) and

M0 = ( P2 + 1)( P2 + 4)/4, M1 = −P2( P2 + 4)/3, M2 = P2( P2 + 1)/12. (116)

The dimensions of the subspaces are T0, 3; T1, 2, ; R1, 2; T2, 1, ; R2, 1.For symmetric second order tensors the projectors M0, M1 and M2 have the same form as in

(116) in terms of P, and may also be expressed in terms of the fundamental vectors introducedearlier. Thus, using (87) and (89),

M0 =1

4(P2 + I)(P2 + 4I) = iit + pdp

td , M1 = u+u∗

+ + u−u∗−, M2 = v+v∗

+ + v−v∗−. (117)

Using these and P1, P2 from (87), it follows that the subspace associated with M0 is two dimen-

sional, while M(1)1 , M

(2)1 , M

(1)2 , and M

(2)2 have dimension one. This Cartan decomposition of the

6-dimensional space Sym agrees with a different approach by Forte and Vianello (8). Alternatively,using (117) and the dyadics in (85), it follows that

M0 = pppp +1

2(I − pp)(I − pp), (118a)

M1 =1

2(pq + qp)(pq + qp) +

1

2(pr + rp)(pr + rp), (118b)

M2 =1

2(qq − rr)(qq − rr) +

1

2(qr + rq)(qr + rq). (118c)

The action of M0 on a second order symmetric tensor is

M0T = T0 = T‖pp + T⊥(I − pp), (119)

where T‖ = T : pp and T⊥ = 12(trT − T : pp). It is clear that T0 is unchanged under rotation

about p. Similarly, M1T transforms as a vector under rotation, hence the suffix 1, while M2T

transforms as a second order tensor with elements confined to the plane orthogonal to the axis ofrotation.

Let p = e3, then using (117)1 or the simpler (118a) yields

T0 = M0(e3)T =

12

0 0 0 0 00 1

20 0 0 0

0 0 1 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

T =

(T1 + T2)/2

(T1 + T2)/2

T3

000

. (120)

22

The projection T0 can be identified as the part of T which displays hexagonal (or transversely

isotropic) symmetry about the axis e3. To be more precise, if we denote the projection M0T asTHex, then THex is invariant under the action of the symmetry group of rotations about p. Itmay also be shown that of all tensors with hexagonal symmetry THex is the closest to T using aEuclidean norm for distance. Similarly,

T1 = M1T =(0, 0, 0, T4, T5, 0

)t, T2 = M2T =

(1

2(T1 − T2),

1

2(T2 − T1), 0, 0, 0, T6

)t, (121)

and using eqs. (82) and (121),

R1 =(0, 0, 0, T5, −T4, 0

)t, R2 =

(− 1√

2T6,

1√2T6, 0, 0, 0,

1√2(T1 − T2)

)t. (122)

It can be checked that under rotation, we have

QT0 = T0, QTj = cos jθTj + sin jθRj , QRj = cos jθRj − sin jθTj, j = 1, 2. (123)

Thus, the singleton T0 and pairs (T1, R1) and (T2, R2) form closed groups under the action of therotation.

6.2.3 Elastic moduli, n=4

The 5-dimensional null space of P is equal to the set of base elasticity tensors for transverse isotropy(23; 8; 14). The projector follows from eqs. (106) and (107) as

M0 ≡1

(4!)2(P2 + 1) (P2 + 4) (P2 + 9) (P2 + 16) =

∑

x=a,b,c,d,e

v0xv0x . (124)

This is equivalent to the 8-th order projection tensor PHex of eq. (68c) of Moakher and Norris (17),who use the base tensors suggested by Walpole (22). The latter representation is useful becauseWalpole’s tensors have a nice algebraic structure, making tensor products simple. Alternatively, inthe spirit of the 21 × 21 representation of elastic moduli, (124)1 and (101) yield

M0(e3) =

MHex 09×12

012×9 012×12

, MHex =

38

38

0 0 0 14√

20 0 1

438

38

0 0 0 14√

20 0 1

4

0 0 1 0 0 0 0 0 00 0 0 1

212

0 0 0 00 0 0 1

212

0 0 0 01

4√

21

4√

20 0 0 3

40 0 − 1

2√

2

0 0 0 0 0 0 12

12

00 0 0 0 0 0 1

212

014

14

0 0 0 − 12√

20 0 1

2

.

(125)

The projection M0 of (125) is identical to the matrix derived by Browaeys and Chevrot (6) forprojection of elastic moduli onto hexagonal symmetry. Specifically, the 9×9 matrix MHex is exactlyM(4) of Browaeys and Chevrot (see Appendix A of (6)). Hence, M0 of (125) provides an algorithmfor projection onto hexagonal symmetry for arbitrary axis p.

The subspace of T0 is 5-dimensional, while the subspaces (T1, R1), (T2, R2), (T3, R3) and (T4, R4)have dimensions 6, 6, 2 and 2, respectively. These may be evaluated for arbitrary axis of rotationp using the prescription in eqs. (28c) and (113).

23

A. Appendix: A recursion

Equation (37) provides an expansion in terms of basic trigonometric functions. An alternativerecursive procedure is derived here. Consider the expression (37) written

eB = fm(B) , (A.1)

where the function fm(B) is a finite series defined by the m distinct pairs of non-zero eigenvaluesof B. Suppose the set of eigenvalues is augmented by one more pair, ±icm+1, with the othersunchanged. Then the sums and products in (37) change to reflect the new eigenvalues,

eB = fm+1(B) = I +m+1∑

j=1

[cj sin cj B + (1 − cos cj) B2

]c−2j

m+1∏

k=1k 6=j

(B2 + c2k

c2k − c2

j

). (A.2)

The effect of the two additional eigenvalues can be isolated using the identity

B2 + c2m+1

c2m+1 − c2

j

= 1 +B2 + c2

j

c2m+1 − c2

j

, (A.3)

to give

fm+1 = fm +

( m+1∑

j=1

cj sin cj B + (1 − cos cj) B2

c2j

m+1∏l=1l 6=j

(c2l − c2

j )

) m∏

k=1

(B2 + c2k) . (A.4)

This defines the sequence, starting with f0 = I.Applying this to the rotation exp(θ P), and noting that the eigenvalues of the skew symmetric

tensor P correspond to cj = jθ, gives

P2n+2(θ, x) = P2n(θ, x) +

( n+1∑

j=1

j sin jθ x + (1 − cos jθ) x2

j2n+1∏l=1l 6=j

(l2 − j2)

) n∏

k=1

(x2 + k2) . (A.5)

The sum over trigonometric functions may be simplified using formulas for sin nθ and cos nθ interms of cos θ/2 powers of sin θ/2 in Section 1.33 of Gradshteyn et al. (10). Thus,

n+1∑

j=1

sin jθ

jn+1∏l=1l 6=j

(l2 − j2)

=(2 sin θ

2)2n+1

(2n + 1)!cos

θ

2, (A.6)

n+1∑

j=1

1 − cos jθ

j2n+1∏l=1l 6=j

(l2 − j2)

=(2 sin θ

2)2n+1

(2n + 1)!

sin θ2

n + 1, (A.7)

and hence,

P2n+2(θ, x) = P2n(θ, x) +(2 sin θ

2)2n+1

(2n + 1)!

(cos

θ

2x + (n + 1)−1 sin

θ

2x2)

n∏

k=1

(x2 + k2) , (A.8)

from which the expression (18b) follows.

24

References

[1] M. Abramowitz and I. Stegun. Handbook of Mathematical Functions with Formulas, Graphs,and Mathematical Tables. Dover, New York, 1974.

[2] G. Andrews. Euler’s ’exemplum memorabile inductionis fallacis’ and q-trinomial coefficients.J. Amer. Math. Soc., 3:653–669, 1990.

[3] B. A. Auld. Acoustic Fields and Waves in Solids, Vol. I. Wiley Interscience, New York, 1973.

[4] O. Bottema and B. Roth. Theoretical Kinematics. North- Holland Publishing Company,Amsterdam, 1979.

[5] Ph. Boulanger and M. Hayes. Bivectors and waves in mechanics and optics. London: Chapmanand Hall, 1993.

[6] J. T. Browaeys and S. Chevrot. Decomposition of the elastic tensor and geophysical applica-tions. Geophys. J. Int., 159:667–678, 2004.

[7] H. Cheng and K. C. Gupta. An historical note on finite rotations. J. Appl. Mech. ASME,56:139–145, 1989.

[8] S. Forte and M. Vianello. Functional bases for transversely isotropic and transverselyhemitropic invariants of elasticity tensors. Q. J. Mech. Appl. Math., 51:543–552, 1998.

[9] J. Gallier and D. Xu. Computing exponentials of skew-symmetric matrices and logarithms oforthogonal matrices. Int. J. Robotics and Automation, 17:1–11, 2002.

[10] I. S. Gradshteyn, I. M. Ryzhik, A. Jeffrey, and D. Zwillinger. Table of Integrals, Series, andProducts. Academic Press: San Diego, CA, 2000.

[11] J. J. Gray. Olinde Rodrigues’ paper of 1840 on transformation groups. Arch. History ExactSci., 21:375 – 385, 1980.

[12] K. Helbig. Representation and approximation of elastic tensors. In E. Fjaer, R. M. Holt,and J. S. Rathore, editors, Seismic Anisotropy, pages 37–75, Tulsa, OK, 1996. Society ofExploration Geophysicists.

[13] R. A. Horn and C. R. Johnson. Topics in Matrix Analysis. Cambridge University Press,Cambridge, UK, 1991.

[14] J. Lu and P. Papadopoulos. Representations of Kronecker powers of orthogonal tensors withapplications to material symmetry. Int. J. Solids Struct., 35:3935–3944, 1998.

[15] M. M. Mehrabadi and S. C. Cowin. Eigentensors of linear anisotropic elastic materials. Q. J.Mech. Appl. Math., 43:15–41, 1990.

[16] M. M. Mehrabadi, S. C. Cowin, and J. Jaric. Six-dimensional orthogonal tensor representationof the rotation about an axis in three dimensions. Int. J. Solids Struct., 32:439–449, 1995.

[17] M. Moakher and A. N. Norris. The closest elastic tensor of arbitrary symmetry to an elasticitytensor of lower symmetry. J. Elasticity, 85:215–263,, 2006.

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[18] F. D. Murnaghan. The Theory of Group Representations. The Johns Hopkins Press, Baltimore,MD, 1938.

[19] A. N. Norris. Optimal orientation of anisotropic solids. Q. J. Mech. Appl. Math., 59:29–53,2006.

[20] P. Podio-Guidugli and E. G. Virga. Transversely isotropic elasticity tensors. Proc. R. Soc. A,A411:85–93, 1987.

[21] O. Rodrigues. Des lois geometriques qui regissent les deplacements d’un systeme solidedans l’espace, et de la variation des coordonees proventant de ces deplacements consideresindependent des causes qui peuvent les produire. J. de Math. Pures et Appliquees, 5:380–440,1840.

[22] L. J. Walpole. The elastic shear moduli of a cubic crystal. J. Phys. D: Appl. Phys., 19:457–462,1986.

[23] Q.-S. Zheng and A. J. M. Spencer. On the canonical representations for Kronecker powersof orthogonal tensors with application to material symmetry problems. Int. J. Engng. Sc.,31:617–635, 1993.

26

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0 0 0 0 0 0 0 0 0 0 0 0 0 2 p2 0 0 0 −2 p3 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 p3 −2 p1 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 2 p1 0 0 0 −2 p2 0 0 0 0

0 0 0 0 0 0 0 0 0 0 −

√

2p2

√

2p3 −

√

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2p1 0 0 0 0 0

0 0 0 0 0 0 0 0 0√

2p1 0 −

√

2p3 0 −

√

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2p2 0 0 0 0

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√

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2p3 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 −2 p1 0 0 2 p1 0 0 0 −

√

2p2

√

2p3

0 0 0 0 0 0 0 0 0 0 0 0 0 −2 p2 0 0 2 p2 0√

2p1 0 −

√

2p3

0 0 0 0 0 0 0 0 0 0 0 0 0 0 −2 p3 0 0 2 p3 −

√

2p1

√

2p2 0

0 0 0 0 −

√

2p1

√

2p1 0 0 0 0 0 0 0 p3 0 0 0 −p2 0 −

√

2p3

√

2p2

0 0 0√

2p2 0 −

√

2p2 0 0 0 0 0 0 0 0 p1 −p3 0 0√

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√

2p1

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√

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2p3 0 0 0 0 0 0 0 p2 0 0 0 −p1 0 −

√

2p2

√

2p1 0

0 0 −2 p1

√

2p1 0 0 2 p1 0 0 0 0 −p2 0 0 0 0 p3 0 0 0 −

√

2p2

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2p2 0 0 2 p2 0 −p3 0 0 0 0 0 0 0 p1 −

√

2p3 0 0

0 −2 p3 0 0 0√

2p3 0 0 2 p3 0 −p1 0 0 0 0 p2 0 0 0 −

√

2p1 0

0 2 p1 0 −

√

2p1 0 0 −2 p1 0 0 0 p3 0 0 0 −p2 0 0 0 0√

2p3 0

0 0 2 p2 0 −

√

2p2 0 0 −2 p2 0 0 0 p1 −p3 0 0 0 0 0 0 0√

2p1

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√

2p3 0 0 −2 p3 p2 0 0 0 −p1 0 0 0 0√

2p2 0 0

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√

2p1

√

2p1 0 −

√

2p3

√

2p2 0√

2p3 0 0 0 −

√

2p2 0 −p3 p2

0 0 0 0 0 0√

2p2 0 −

√

2p2

√

2p3 0 −

√

2p1 0 0√

2p1 −

√

2p3 0 0 p3 0 −p1

0 0 0 0 0 0 −

√

2p3

√

2p3 0 −

√

2p2

√

2p1 0√

2p2 0 0 0 −

√

2p1 0 −p2 p1 0

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A

Table 1. The matrix P(p) ∈ so(21) for 4th order elasticity tensors represented as 21-vectors. The rotation Q = exp(θP) is

given by the 8-th order polynomial Q = P8(θ, P) of eq. (20).